Graph-based determination of initial-synchronization beam scanning

ABSTRACT

Techniques for determining beam-sweeping patterns for synchronization signals transmitted in a region by several access nodes in a network, where each access node is connected to a corresponding array of antenna elements. An example method includes modeling a total power function for the power transmitted in the synchronization signals, as a factor graph having a plurality of check nodes and variable nodes, each check node corresponding to a virtual wireless device in the region and each variable node corresponding to an available beam for an access node. The virtual wireless devices are emulated so as to implement quality-of-service constraints on synchronization signals received by the virtual wireless devices. An iterative message-passing algorithm, such as a min-sum algorithm, is applied to the modeled total power function, to determine a sequence of power levels, for each access node, for sweeping synchronization signal beams, so as to minimize the total power function.

TECHNICAL FIELD

The presently disclosed techniques and apparatus are generally relatedto wireless communications networks, and are more particularly relatedto techniques and apparatus for determining scanning beam patterns to beused by access nodes transmitting synchronization signals in a region,to be used by wireless devices in obtaining initial synchronization tothe access nodes for accessing communications services.

BACKGROUND

The millimeter-wave portion of the radio spectrum, e.g., includingfrequencies in the range of 60 GHz or so, is expected to provide manybenefits for fifth-generation (5G) wireless communications systems. Theaccessible channel bandwidths in the millimeter-wave band arepotentially larger than those available to commercial wireless systemsoperating at lower bands, such as those systems using the currentLong-Term Evolution (LTE) radio access technologies developed by themembership of the 3^(rd)-Generation Partnership Project (3GPP). Further,the smaller radio wavelengths in the millimeter-wave band allow theradio transceivers to have more compact hardware when supportingmultiple antennas, due to the fact that the antenna element separationin a multi-antenna scenario is typically proportional to the wavelength.

One implication of this latter advantage of the millimeter-wave band isthat a large number of antenna elements can be connected to radio accessnodes (hereinafter referred to as simply access nodes, or ANs) whilemaintaining relatively “regular” sizes, compared to previously deployednodes. These antenna elements, which may be arranged as arrays ofhorizontally and/or vertically spaced elements, may be used to formnarrow beams, concentrating, for example, all the access node's transmitpower in a specific/desired direction. This way, distant wirelessdevices (commonly called user equipments or UEs in 3GPP specificationdocuments) can be reached without causing high interference to devicesother than those specifically targeted by the access nodes.

On the other hand, initial synchronization of UEs in a millimeter-wavewireless network, which is generally the first step taken by UEs whenattempting to access the network for services, raises some challengingissues. In this context, a key challenge is ensuring that whenever a newUE tries to join such a network, at least one AN in the vicinity of theUE should be able to provide it with signal quality that is good enoughto establish a connection between the AN and the UE. At the same time,however, as little power as possible should be devoted to signalstransmitted to the UE, so as to allow for support of many other UEs, tominimize interference to other devices, and to minimize overall powerconsumption of the network. However, the use of narrow beams for initialsynchronization purposes provides good signal quality only to a smallfraction of the area to be covered. If a new UE arrives in apoorly-covered area, one consequence of using narrow beams for thetransmission of synchronization signals can be that this UE is unable tojoin the network, since it cannot listen to and decode satisfactorilyany synchronization signal from the ANs.

Inactivating antenna elements in the array of antenna elements availableto a given access node makes it possible to create wider beam patterns,with the limit being reached by transmission from only a single antennaelement. In some possible configurations, however, such asconfigurations in which there is one power amplifier per antennaelement, this may entail a reduction in total conducted power into thearray. It is therefore more power-efficient, as a general matter, to useall of the available antenna elements to radiate power.

Assuming the use by each access node in a given region of all theantenna elements available to it, it is therefore desirable to minimizethe total consumed power by adjusting the power level of every beam,from every access node, in each transmit-time interval, so that every UEin the region can perform initial synchronization. This leads to abeam-sweep procedure, in which narrow beams, one at each AN, aresimultaneously transmitted in contiguous transmit-time intervals, namelybeam sweep instances, in order to radiate energy over the area where UEsmay appear and try to establish connection, until the whole area isscanned.

To minimize the total consumed power for this initial synchronizationprocess, an optimization problem can be formulated where the objectiveis to minimize the sum of all the transmit power levels to be usedduring the beam scan. The objective is subject to a set ofquality-of-service (QoS) constraints (e.g., minimumsignal-to-interference-plus-noise ratio (SINR) requirements for thereception of synchronization signals by UEs in the region) and a set oftransmit power constraints. The result is the feasible power settings,for all transmitted synchronization beams, where the element-wise sum ofthe power settings is minimized. From the resulting power settings, thebeam sweep pattern to be used in the beam scan is derived. Ideally, thetransmission of synchronization signals using the resulting powersettings provides good signal quality to any new UE in the area that maytry to synchronize to an AN.

By its nature, the problem described above is combinatorial, with highcomputational complexity for large-scale networks. Thus, its optimalsolution is difficult to obtain, as the exhaustive search for theoptimal solution grows exponentially with the number of QoS constraints.Although the combinatorial problem can be transformed into amixed-integer linear program to be solved in a more efficient manner, itis still intractable for very-large-scale networks.

One possible low-complexity approach to defining beam-sweeping patternsfor the ANS is to choose the beam sweep pattern, i.e., the sequence ofbeams that ANs follow over beam sweep instances, at random, with nopower optimization. This is the approach described in C. N. Barati, S.A. Hosseini, S. Rangan, P. Liu, T. Korakis, and S. S. Panwar,“Directional cell search for millimeter wave cellular systems,” in IEEE15th International Workshop on Signal Processing Advances in WirelessCommunications (SPAWC), June 2014]. However, this approach would makeANs transmit synchronization signals to areas where UEs never (orrarely) appear. Thus, unnecessary power is consumed. In addition, UEsmay experience high interference levels from the randomly swept beams.

Another approach is to design the beam pattern and optimize the powersettings based on historical statistics for UEs, as collected by thewireless network. This is the approach described in I. M. Guerreiro, J.Axnäs, D. Hui, and C. C. Cavalcante, “Power-efficient beam sweeping forinitial synchronization in mm-Wave wireless networks,” in IEEE 16^(th)International Workshop on Signal Processing Advances in WirelessCommunications (SPAWC), June 2015. From such a data set, ANssub-optimally find a power setting sufficient to provide goodsynchronization signal quality for UEs to synchronize. As a consequence,the beam sweep pattern is derived from power settings, where onlydirections associated with non-zero power levels are used. However, thespecific algorithms detailed in the paper cited above are performed in acentralized manner, and provide solutions far from the optimum.

SUMMARY

According to some of the techniques detailed herein, ANs exchangemessages, e.g., via a backhaul, to jointly find both an optimizedtransmit power setting and an optimized beam sweep pattern for each ofthe ANs. The messages to be passed define a protocol to be followed byANs so that a message-passing algorithm is properly performed. Thealgorithm can be carried out offline in a decentralized manner, anddepends on the availability of historical statistics collected by thesystem. Advantages of some embodiments of the described solutionsinclude that the radio channel is used less often with the optimizedbeam scan pattern. Energy is also saved, and interference reduced, dueto the optimized power setting. According to some of these embodiments,as described further below, messages are calculated locally at ANs basedon historical statistics provided to ANs by the system.

An example method, carried out in one or more nodes of a wirelesscommunications system, is for jointly determining beam-sweeping patternsfor synchronization signals transmitted in a region by each of aplurality of access nodes in a wireless network, where each access nodeis connected to a corresponding array of horizontally and/or verticallyspaced antenna elements and is configured to sweep a synchronizationsignal in a node-specific beam sequence, using the corresponding array,where, for each access node, the node-specific beam sequence is definedby a sequence of power levels corresponding to distinct beam anglesavailable to the access node. This example method includes modeling atotal power function corresponding to a total power transmitted in thesynchronization signals by the plurality of access nodes, for a givenperiod, wherein said modeling of the total power function comprisesmodeling the total power function as a factor graph having a pluralityof check nodes and variable nodes, each of the check nodes correspondingto one of a plurality of emulated virtual wireless devices in the regionand each of the variable nodes corresponding to one of the access nodesand to one of the beam angles available to the one of the access nodes.The method further includes emulating the plurality of virtual wirelessdevices so as to implement quality-of-service constraints onsynchronization signals received by the virtual wireless devices, andapplying an iterative message-passing algorithm to the modeled totalpower function, to determine the sequence of power levels for each ofthe plurality of access nodes, so as to minimize the total powerfunction, subject to one or more iteration-stopping criteria.

Corresponding systems, which may include one or more access nodesconfigured to carry out, in a distributed manner, the method summarizedabove, are also described, as are corresponding computer programproducts. It will be appreciated that the disclosed techniques,apparatus, and systems further include variations of those summarizedabove, as detailed below and as illustrated in the attached figures.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 illustrates an example network scenario in which the presenttechniques may be applied.

FIG. 2 illustrates an example of a factor graph with check nodes andvariable nodes, where the check nodes and variable nodes are mapped tovirtual users (wireless devices) and access nodes, respectively.

FIG. 3 illustrates an example of a fully-connected factor graph withthree check nodes and three variable nodes.

FIG. 4 is a process flow diagram illustrating an example methodaccording to some embodiments of the presently disclosed techniques.

FIG. 5 illustrates a regular factor graph with undirected edges and anequivalent factor graph with node replication.

FIG. 6 illustrates an equivalent factor graph of the fully-connectedfactor graph example in FIG. 3.

FIG. 7 is a process flow diagram illustrating an example methodaccording to some embodiments of the presently disclosed techniques.

FIG. 8 is a block diagram illustrating an example access node.

DETAILED DESCRIPTION

As discussed above, to minimize the total consumed power during initialsynchronization, in a system where multiple access nodes are sweepingnarrow beams in an area, an optimization problem can be formulated suchthat the objective is to minimize the sum of all the transmit powerlevels to be used during the beam scan. As noted above, the objective issubject to a set of QoS constraints and a set of transmit powerconstraints. The result is the power settings for the swept beams thatminimize the total power consumption.

Developing the set of QoS constraints depends on the availability ofinformation characterizing the reception of synchronization signals atvarious points in the region covered by the swept beams. Thisinformation may be derived from a set of historical statistics of users.That is, synchronized UEs report, to the network, some informationcharacterizing the received signal quality during synchronization. Eachsuch report can then be treated as a “virtual UE” that emulates a UE inthe optimization problem, with the SINR for each UE being a function ofthe spatial directions in which the synchronization signals aretransmitted.

Based on these virtual UEs, the optimization problem described above canbe solved using a decentralized approach in which ANs exchange messages,e.g., via a backhaul, to jointly find both optimized transmit powersettings and an optimized beam sweep pattern for each AN. The messagesto be passed define a protocol to be followed by ANs, so that amessage-passing algorithm can be executed. This message-passingalgorithm can be carried out offline, in a decentralized manner, giventhe availability of historical statistics collected by the system.

Accordingly, the techniques described herein provide a method to computean allocation of a set of resources to be used by access nodes duringinitial synchronization of users in a network. An example of this set ofresources is a set of power levels, namely power settings, ofsynchronization signal beams transmitted from ANs to users that may betrying to establish connection to the network. These power settings,which corresponding to particular beam azimuths and elevations, define abeam sweep pattern, where beams associated with non-zero power levelsare transmitted following a pre-defined sequence. Another example of anallocation of a set of resources that may be computed according to theprocedures generally described herein is to determine the time and/orbeam-width for the transmitted synchronization signal beams, instead ofthe power levels.

This method can be performed in an “offline” manner, that is, before thetransmission of synchronization signals according to the calculatedconfiguration. As an example, the system provides a set of historicalstatistics for wireless devices to the access nodes in a given region.This historical data reflects measurements, link quality information,and the like, as previously reported by wireless devices aftersynchronization. The type of data reported by the wireless devicescomprises the received signal power and transmit beam identification,for example, where the transmit beam identification identifies aspecific beam (and thus indicates the source of the beam as well as itsspecific azimuth and/or elevation). Thus, an AN (or other node analyzingthe reported data) can know, for a given beam direction, what thereceived signal qualities were at the UEs that reported such data. Thecollected data might be arranged, for example, into a large array whoserows correspond to different beams m and whose columns correspond todifferent UEs k, with each entry (m, k) being a corresponding receivedsignal quality.

Note that initially, when no such historical user statistics areavailable to the system, each AN may transmit at full power as it sweepsover all possible beams, while it gradually builds up the collection ofhistorical statistics.

The techniques described herein can also be performed in an “online”manner. As an example, access nodes in operation receive data from usersafter synchronization and process such data to update their resourcesand beam pattern in a more or less continuous manner.

In some embodiments, the methods described herein are run periodically.As an example, the set of historical statistics is updated or more datais added by the system, e.g., as, more wireless devices establishconnection and report data to the system. Then, the allocation ofresources, e.g., the power settings for the available synchronizationbeams, must be updated.

In some embodiments, each run of the method comprises the execution of adecentralized algorithm performed by access nodes in the network. As anexample, each access node computes power levels to be used duringsynchronization signal transmissions, based on the set of historicaldata and also on information received from neighboring access nodesthrough a backhaul. This decentralized approach is based onmessage-passing. As an example, a message-passing algorithm is developedto be applied to the problem of initial synchronization via beam scan.Access nodes exchange messages that convey information indicating howeach power level value per beam impacts the total consumed power. Themessage-passing can be based on the sum-product algorithm, for example,as described in F. R. Kschischang, B. J. Frey, H.-A. Loeliger, “Factorgraphs and the sum-product algorithm,” in IEEE Transactions onInformation Theory, vol. 47, no. 2, February 2001, but is not limited tothis particular approach.

Alternatively, the techniques can be run in a centralized manner, whilestill using a message-passing algorithm. In this case, the messages,detailed examples of which are described below, are “passed” betweenemulated access nodes, and between the emulated access nodes andemulated virtual wireless devices. As an example of this approach, acentral processing unit having network connections to access nodesgathers the set of historical statistics of users, to jointly computethe set of resources to be applied by the access nodes. Then, thecomputed set of resources is sent to access nodes to be used for thetransmissions of the synchronization signal beams.\

System Model

As an example to describe the system model, consider an indoormillimeter-wave scenario where N ANs are arbitrarily placed to providean adequate coverage to K UEs for initial synchronization. Let N be theset of all the ANs in the network, and let K be the set of virtual UEsavailable in the historical data observations collected over time. TheseUE records may denote the received signal power per beam observed andreported by UEs. From those records, relevant areas where UEs are mostlikely to arrive and request connectivity can be estimated.

In the following, it is assumed each AN is connected to a large numberof M×M antenna elements, vertical and horizontally spaced by d, throughthe use of 2-dimensional uniform planar phased antenna arrays. Moreover,for simplicity, each UE is assumed to be (but is not limited to be) asingle-antenna receiver, which ideally receives signalsomni-directionally. The use of a massive number of transmit antennas(e.g., 64 antenna elements) at each AN results in a particular beamshape that has a main lobe with narrow beam width and high antenna gain.The direction in which this main lobe is pointed depends on what valuethe relative phase excitation between adjacent antenna elements takeson. Note that the techniques described here are not limited to theseantenna configurations, but can be used for systems where ANs havedifferent numbers of antennas elements, and, for example, where at leastsome of the antenna arrays may be exclusively horizontal or vertical.

FIG. 1 is a simplified illustration of a region 150 that includesseveral ANs 110 and several UEs 120. The ANs 110 are illustrated astransmitting narrow-beam synchronization signals; while multiple beamsare illustrated, it will be appreciated that each AN 110 may betransmitting only one beam at any given instance. In simple terms, theoptimization problem discussed herein involves determining theparticular beams to be transmitted by each AN, as well as the powerlevels associated with those beams, to ensure that all of the UEs 120receive synchronization signal transmissions that meet their respectiveQoS constraints, e.g., that provide an appropriate SINR for thesynchronization signal as received by each UE 120.

Beam Sweep Code Book

Each of the ANs in the system described above is able to sweep (or scan)its surroundings by varying the azimuth and elevation angles associatedwith its antenna array. In principle, the azimuth and elevation anglescan take on any value in [0 2π], each pair of values defining then asteering vector. The mapping between those angles and the steeringvector can be either a continuous or a discrete function. In thediscrete case, the sets of azimuth and elevation angles are finite andpre-defined, where each ordered pair of angles defines a beam/direction.Also, all the ANs simultaneously transmit beams, but only one beam perAN is transmitted in a given beam sweep instance. One by one, beams aresequentially transmitted at each AN and eventually the entire relevantarea is swept and properly covered.

As an example of a pre-defined discrete set (codebook) of beams, let

be an index set enumerating the beams available at each AN. That is,

={1,2, . . . ,L}where L is the number of available beams. Also, let u_(n,l) denote aunit-norm weight vector that represents the (narrow) beam l∈

transmitted through the M×M transmit antenna elements at AN n. Assumingtransmit phased arrays, each weight vector u_(n,l) is then defined asfollows:

${u_{n,l} = {\frac{1}{M}\begin{bmatrix}e^{{- \frac{j\; 2\;\pi}{\lambda}}{({x_{n,\; 1} - x_{n,0}})}^{T}a_{l}} \\e^{{- \frac{j\; 2\;\pi}{\lambda}}{({x_{n,\; 2} - x_{n,0}})}^{T}a_{l}} \\\vdots \\e^{{- \frac{j\; 2\;\pi}{\lambda}}{({x_{n,M^{2}} - x_{n,0}})}^{T}a_{l}}\end{bmatrix}}},$where:a_(l)=[cos θ_(l) sin ϕ_(l) sin θ_(l) sin ϕ_(l) cos ϕ_(l)]^(T),and where vectors x_(n,m) and x_(n,0) stand for the column vectorscollecting 3-dimensional Cartesian coordinates of antenna element m atAN n and the reference point (i.e., the center point of the antennaarray) of AN n, respectively, λ denotes the wavelength, and θ_(l) andϕ_(l) are the azimuth and elevation angles that specify the relativephase excitation between antenna elements of AN n.

Furthermore, letw _(n,l)=√{square root over (P _(n,l))}u _(n,l)represent the precoding weight (steering) vector where P_(n,l) denotesthe transmit power that AN n sets to transmit its beam. Thus, the beamsweep instance l is defined as the transmit-time interval when AN ntransmits beam l with power P_(n,l), for all n∈

.

For simplicity, beams are ordered as a linear sequence so that each beamis indexed by the index m. Thus, the ordered pair (n, l), which refersto the AN n∈

and its beam l∈

, is mapped as (n, l)→m, so thatm=L(n−1)+l,and, reversely, for m=1, . . . , NL,

${n = \left\lceil \frac{m}{L} \right\rceil},$l=mod [m−1,L]+1,

where ┌⋅┐ and mod [⋅,⋅] stand for the ceiling function and the modulooperation, respectively. Then, a total of NL beams makes up the set

={1,2, . . . ,NL}.Note that this approach eases the representation of beams through theirassociated power levels in a factor graph. Accordingly, each power levelP_(n,l) can be mapped into P_(m). The same is applied to weight vectors,i.e., each u_(n,l) is mapped into u_(m) and each w_(n,l) is mapped intow_(m). Eventually, when a UE is assigned to a beam indexed by m, it caneasily map m back into (n, l).

Neighborhood Definition and Performance Metric

For each virtual UE k, let

be an index set of beams that virtual UE k is capable of listening toduring synchronization and reporting data for. For each beam m, let

be an index set of virtual UEs that the system is capable of receivingdata from, regarding beam m as transmitted from its associated AN.

For each beam m, an individual power level P_(m), subject to a maximumpower constraint, i.e. P_(max), is calculated, so that there exists atleast one precoding weight vector w_(m) that provides goodsynchronization signal quality for all the virtual UEs in

. Let G_(m,k) denote the equivalent channel gain between virtual UE kand AN transmitting beam m. The SINR observed by virtual UE k listeningto beam m is then defined as

${\Gamma_{m,k} = \frac{P_{m}G_{m,k}}{{{P_{m^{\prime}}G_{m^{\prime},k}} + \sigma_{k}^{2}}\;}},$where

is the set of beams that interfere beam mat virtual UE k, defined as

={m′∈

|mod [|m′−m|,L]=0,m′≠m}.The interfering beams of beam m are those that are transmitted in thesame beam sweep instance by neighboring ANs.

To deal with the SINR constraints, an indicator function 1_(k) isdefined as

${1_{k}\left\lbrack {\Gamma_{m,k} \geq \gamma} \right\rbrack} = \left\{ {\begin{matrix}0 & {{\exists m},{\Gamma_{m,k} \geq \gamma},} \\{+ \infty} & {otherwise}\end{matrix},} \right.$where γ is a SINR threshold above which every Γ_(m,k) must be, forsuccessful synchronization and robust decoding of synchronizationsignals. The indicator function penalizes any infeasible power settingby returning +∞ (or other very large value).

A power consumption model function Ψ_(m) is introduced to account forthe relation between the transmit power and the consumed power at eachAN. In general, it can have any shape to take into accountnonlinearities and power dissipation, but it can be simply a linearrelation asΨ_(m)(P _(m))=P _(m),which means that the power consumption depends only on the transmitpower.

Another aspect is the fact that some a priori information ANs may obtainfrom the historical statistics can be incorporated into each Ψ_(m). Asan example, virtual UEs may inform the system the power levels that ANsshould transmit with in order to satisfy every SINR constraint. Let P_(m,k) denote the minimum power level that beam m should be transmittedwith in order to satisfy the SINR constraint of virtual UE k, defined as

${{\overset{\_}{P}}_{m,k} = {\frac{\gamma}{G_{m,k}}\left( {{\sum\limits_{j \in \mathcal{M}_{m,k}}{P_{j}G_{j,k}}} + \sigma_{k}^{2}} \right)}},{\forall{m \in}}$Note that each P _(m,k) depends on its set of interfering beams

. If such information is taken into account, then the search space ofthe best power setting becomes a function of every P _(m,k). In otherwords, ANs do not have any incentive to transmit a beam with any powerdifferent from the minimum power levels P _(m,k). Consequently, eachfunction Ψ_(m) looks like a sum of step functions, each step being afunction of an individual minimum power level.

Optimization Problem Formulation:

The total consumed power function ƒ in the network is formulated as anobjective along with a penalty factor represented by the indicatorfunction, as follows:

$\begin{matrix}{{f\left( \left\{ P_{m} \right\}_{m \in \mathcal{M}} \right)} = {{\sum\limits_{m \in \mathcal{M}}\Psi_{m}} + {1_{k}\left\lbrack {\Gamma_{m,k} \geq \gamma} \right\rbrack}}} \\{= {{\omega_{j,k}\Psi_{j}} + {1_{k}\left\lbrack {\Gamma_{m,k} \geq \gamma} \right\rbrack}}} \\{{= {\left( {{\omega_{j,k}\Psi_{j}} + {1_{k}\left\lbrack {\Gamma_{m,k} \geq \gamma} \right\rbrack}} \right)}},}\end{matrix}$where ω_(j,k) denotes a weight to avoid double counting of variables,and is defined so that

ω_(j, k) = 1, ∀j ∈ ℳ.

Note that ƒ factorizes into a sum of K terms (factors), and each set

determines the interdependency of these factors. Note that the set ofhistorical statistics is assumed to be rich enough so that every beam in

has at least one UE in

that can listen to it. Alternatively, the set

can be replaced with some set

defined as the finite union of sets

, for all k∈

.

Finally, the problem of minimizing the total consumed power in thenetwork can then be stated as

$\min\limits_{{\{{P_{m} \in {\lbrack{0P_{\max}}\rbrack}}\}}_{m \in \mathcal{M}}}{\sum\limits_{k \in \mathcal{K}}{\left( {{\sum\limits_{j \in \mathcal{A}_{k}}{\omega_{j,k}{\Psi_{j}\left( P_{j} \right)}}} + {1_{k}\left\lbrack {\Gamma_{m,k} \geq \gamma} \right\rbrack}} \right).}}$

General Graph-based Framework

Due to the structure of the equation above, a message-passing algorithmframework, such as the framework of the min-sum algorithm, can beapplied to find the power setting that minimizes the total consumedpower in the network. Note that while the use of the min-sum algorithmis described herein, the optimization problem described above may beformulated in different but substantially equivalent ways, such thatother algorithms, such as the max-sum algorithm (e.g., where thenegative of the total power function is optimized) or min-productalgorithm (e.g., where the total power function is in the log domain)may be used, instead of the min-sum algorithm. It will be appreciatedthat the basic approach described here will apply to any of thesealternate formulations of the optimization problem.

The total consumed power function ƒ described above can be graphicallymodeled by a bipartite graph, namely a factor graph, with check nodesand variable nodes. Each check node according to this approachrepresents a virtual UE, while each variable node represents a beam.More precisely, each variable node corresponds to a particular beamdirection, at a particular node, where the transmit power level for thebeam is the “variable.” The check nodes act as entities that checkwhether the SINR constraints of the virtual UEs are satisfied, while thevariable nodes act to establish the transmit power levels of theirrespective beams. FIG. 2 shows an example of factor graph modeling anetwork with L beams per AN, with a total of N ANs and K virtual UEs,which means that there are NL variable nodes and K check nodes. In FIG.2, check nodes are represented as squares and variable nodes as circles.Each access node has L beams and there are N access nodes in thenetwork. A virtual user, represented by check node M_(k), corresponds toa user that synchronized and reported data to the system previously. Inaddition, each variable node represents a beam, where the power levelfor the beam is the modeled variable. Check nodes and variable nodes areconnected through edges. If a variable node is connected to a checknode, it means that the underlying virtual user can detect and decodethe synchronization signal from that beam, at some level. The task ofthe optimization problem is to determine the minimum non-zero powerlevel that should be used by the access node to transmit that beam sothat the virtual user (which emulates an actual device) can detect anddecode the transmitted synchronization signal. This minimum non-zeropower level is the power level that satisfies the quality-of-serviceconstraint for the virtual user—this may be a constraint that the SINRfor the received synchronization level is greater than a threshold γ,for example.

Now, let α_(k→m), referred to herein as a “summary message,” denote themessage to be passed from check node k to variable node m, and letβ_(m→k), referred to herein as an “aggregate message,” denote thenormalized message to be passed from variable node m to check node k.The min-sum algorithm simply iterates between the following two kinds ofmessage computation and exchanges:

-   -   Summary message, from check node to variable node:

${{\alpha_{k\rightarrow m}\left( P_{m} \right)} = {\left\{ {{1_{k}\left\lbrack {\Gamma_{m,k} \geq \gamma} \right\rbrack} + {\sum\limits_{j \in {\mathcal{A}_{k} \smallsetminus {\{ m\}}}}{\beta_{j\rightarrow k}\left( P_{j} \right)}}} \right\}}},$

-   -   where the minimization is performed over all the neighboring        variable nodes except for the destination. Thus, each summary        message is a univariate function of P_(m). Note that Γ_(m,k)        also depends on P_(m).    -   Aggregate message, from variable node to check node:

${{\beta_{m\rightarrow k}\left( P_{m} \right)} = {{\Psi_{m}\left( P_{m} \right)} + {\sum\limits_{j \in {\mathcal{B}_{k} \smallsetminus {\{ k\}}}}{\alpha_{j\rightarrow m}\left( P_{m} \right)}} - \left( {\min\limits_{P_{m}}{\sum\limits_{j \in {\mathcal{B}_{m} \smallsetminus {\{ k\}}}}{\alpha_{j\rightarrow m}\left( P_{m} \right)}}} \right)}},$

-   -   where the sum is performed over all the neighboring check nodes        except for the destination. The third term on right-hand side of        the equation above is a normalizing factor to prevent messages        from increasing endlessly.

Note that the expression of aggregate message computation above containsexplicitly the function Ψ_(m) as an additive term, which depends only onP_(m). Thus, alternatively, a factor node can be introduced and attachedto each variable node to represent each function Ψ_(m). In this case,each additive term Ψ_(m) is in turn moved from the aggregate messageexpression to each new factor node as a factor. Consequently, theresulting factor graph would contain both factor and check nodes, alongwith the variable nodes, as shown in FIG. 3, which shows an example of afully-connected factor graph with three check nodes, three factor nodes,and three variable nodes. Every check node is connected to all thevariable nodes, which means that, in this example, every virtual UEheard all the beams used in the network. In contrast, each factor nodeis connected only to its associated variable node. Each factor node'srole is to pass its factor to its associated variable node.

Granular Message-passing: A Low-complexity Approach

The computational complexity to compute the summary messages and theindividual minimum power levels, as described above, is still high. Eachcheck node must hypothesize every combination of interfering terms.

To decrease complexity, the interference terms can be neglected, sincenarrow beams can be assumed to cause very low interference to oneanother, and since the SINR threshold γ is usually small (e.g., −10 dB)for robust decoding and detection.

Also, note that in general, a user does not need to be served by morethan one beam. Thus, a message from a virtual UE k to a beam m can bedefined as a piecewise function. A threshold is then defined as theminimum power level needed for that beam m to serve virtual UE k, i.e.,P _(m,k). Above such a threshold, the power of the other neighboringbeams can be hypothesized so that they consume as little power aspossible. Otherwise, virtual UE k hypothesizes the minimum powerconsumed by the other beams if at most one of them would serve.

More specifically, summary messages are redefined, according to thislow-complexity approach, by disregarding interference terms, as

${\alpha_{k\rightarrow m}\left( P_{m} \right)} = \left\{ {\begin{matrix}0 & \; & {{{{if}\mspace{20mu} P_{m}} \geq {\overset{\_}{P}}_{m,k}},} \\\min\limits_{j \in {\mathcal{A}_{k} \smallsetminus {\{ m\}}}} & \left( {\min\limits_{P_{j} \geq P_{j,k}}{\beta_{j\rightarrow k}\left( P_{j} \right)}} \right) & {otherwise}\end{matrix},{{{where}{\overset{\_}{P}}_{m,k}} = \frac{{\gamma\sigma}_{k}^{2}}{G_{m,k}}},{\forall{m \in \mathcal{M}}},{k \in {\mathcal{B}_{m}.}}} \right.$Note that the minimum power levels are constant, due to the neglectingof interference. In addition, recall that aggregate messages arenormalized, i.e., their minimum values equal zero.

The expressions above can be understood as answering the question thatcheck node k needs to resolve, which is: “which neighboring beam cansatisfy me with minimum power consumption?” The check node tries toexpress the best answer, per neighboring variable node, through thecomputation and delivery of summary messages. Thus, for a given summarymessage, say α_(k→m)(P_(m)), check node k determines that beam m cansatisfy it with beam powers P_(m) that fall within P_(m,k)≤P_(m)≤P_(max). For this range of beam powers, the summary messagefor beam m returns a value of 0. Note that from the sum of all the mostrecently received aggregate messages other than the one from thetargeted variable node m, the minimum power consumed is zero becauseevery aggregate message is normalized. For beam powers in the rangeP_(m)≤P _(m,k), however, beam m cannot satisfy the check node's QoSconstraint. In this range, some other neighboring beam must satisfy it.To find the beam j∈

\{m} that satisfy it with minimum power consumption, check node kcalculates the minimum values of every message β_(j→k)(P_(j)) within P_(j,k)≤P_(j)≤P_(max) (the inner minimization in the expression above)and choose the minimum among these minimum values (the outerminimization). Check node k is then effectively saying to variable nodem that “for the range P_(m)<P _(m,k), you cannot satisfy me. So, anotherbeam will satisfy my QoS constraints, and this is the resulting power tobe consumed.”

The expression of summary messages above has a low computationalcomplexity. Each check node k has to do only a small number of checks tocompute its summary messages. Furthermore, for binary power levels, allsummary and aggregate messages can be single-step, independently of thenumber of virtual UEs, which decreases even more the complexity ofmessage computation.

Message-passing Algorithm

FIG. 4 is a flowchart illustrating the steps of a message-passingalgorithm. The message-passing algorithm begins, whether or not thelow-complexity approach described above is used, with variable node mcomputing outgoing aggregate message β_(m→k) to check node k, for k∈

, ∀m∈

, with an initialization of all the incoming summary messages fromneighboring check nodes to zero. That is, the initially computedaggregate messages simply equal their respective power consumption modelfunctions. This is shown at block 410. As shown at block 420, theinitially computed aggregate messages are sent by the variable nodes totheir connected check nodes.

Upon receipt of the aggregate messages, as shown at block 430, eachcheck node k computes outgoing summary message α_(k→m) to variable nodem, for m∈

, ∀k∈

, and the computed summary messages are then sent to variable nodes, asshown at block 440. This back-and-forth message exchange defines aniteration of the message passing algorithm. At the end of amessage-passing iteration, e.g., after receiving the summary messagesfrom its connected check nodes, as shown at block 450, each variablenode can compute its corresponding univariate function and thendetermine its power level that minimizes it by

${P_{m}^{*} = {\arg\;{\min\limits_{P_{m}}\left\{ {{\Psi_{m}\left( P_{m} \right)} + {\sum\limits_{j \in \mathcal{B}_{m}}{\alpha_{j\rightarrow m}\left( P_{m} \right)}}} \right\}}}},{m \in {\mathcal{M}.}}$This computation is shown at block 460. The algorithm then iteratesuntil a stopping criterion is reached, as indicated at block 470, afterwhich the final power settings for each access node are computed, asshown at block 480 This criterion may be for example, a predeterminedmaximum number of iterations, or when the power settings computed at theend of each iteration converge to a fixed state.

If the stopping criterion is not reached, as indicated by the “no” arrowleaving block 470, the variable computes aggregate messages again, asshown at block 490, based on the most recent summary messages, and theiteration continues, as shown in the figure.

Emulation of Virtual UEs and Message Exchange Among ANs

Virtual UEs are emulated by ANs to act as users in the system with QoSconstraints to be satisfied during initial synchronization. A givenvirtual UE must then be emulated by at least one AN, e.g., one of theANs for which that virtual UE reported data.

The set

of virtual UEs can be split into subsets

so that

=U_(n∈)

K_(n). Then, for every n∈

, virtual UEs that belong to subset K_(n) are emulated by AN n. Thesubsets K_(n) can be disjoint, which means that every virtual UE isemulated by no more than one AN. However, different virtual UEs canstill be emulated by different ANs. In this case, check nodes in thefactor graph in FIG. 2 are allocated to the ANs to form the subsetsK_(n) at each AN. For example, each virtual UE can be emulated by the ANwith the lowest path loss to itself. If a check node is connected tovariable nodes at different ANs, summary messages are then exchangedamong those ANs, which makes the message exchange among ANs compriseboth summary and aggregate messages.

Alternatively, each check node k may be replicated at all of itsneighboring variable nodes in A_(k). Due to node replication, arrowlessundirected (or bi-directed) edges in the graph are replaced with (uni-)directed edges with arrows. The arrow of a directed edge indicates thedirection in which messages along that edge are passed. FIG. 5illustrates a regular factor graph, on the left-hand side of the figure,as well as its equivalent version with replicated check nodes, on theright-hand side. In the illustration on the right-hand side, undirectededges are replaced with directed edges whose arrows indicate thedirections messages are passed in. The regular factor graph on theleft-hand side of FIG. 5 is then equivalent to the factor graph withreplicated nodes on the right-hand side. However, in the illustration onthe right-hand side of FIG. 5, each pair of variable node and replicatedcheck node in a curved dashed rectangle represents a beam in a given AN.Note that only aggregate messages are passed among beams.

Now suppose that variable nodes are connected to multiple check nodes.For example, in a fully-connected factor graph with three check nodesand three variable nodes, every check node is connected to all thevariable nodes, as was shown in FIG. 3. Then, each AN emulates thevirtual UEs needed for its beams to pass messages also by using nodereplication. Consequently, an equivalent factor graph with replicatedcheck nodes can be found.

FIG. 6 illustrates the equivalent factor graph of the fully-connectedfactor graph example of FIG. 3. Each curved rectangle represents anaccess node. In this example, access node 1 has beams a and b, whileaccess node 2 has beam c. The check nodes inside an access node denotethe emulation of the virtual UEs the beams of that access node wereheard by. Note that the message exchange between beams of the sameaccess node is performed locally, whereas the message exchange amongbeams of different access nodes takes place over some backhaul.

Message passing from variable nodes to check nodes within the same ANare passed locally, which in principle does not cause any impact on thesignaling load over some backhaul that interconnects ANs. On the otherhand, messages from variable nodes to check nodes belonging to differentaccess nodes are sent over the backhaul. Summary message computationassociated with a given check node must be parallelized over ANs relatedto neighboring variable nodes of that check node. That is because eachof these ANs has a replica of that check node.

With the above detailed explanation in mind, it will be appreciated thatFIG. 7 illustrates a generalized method of jointly determiningbeam-sweeping patterns for synchronization signals transmitted in aregion by each of a plurality of access nodes in a wireless network,wherein each access node is connected to a corresponding array ofhorizontally and/or vertically spaced antenna elements and is configuredto sweep a synchronization signal in a node-specific beam sequence,using the corresponding array. For each access node, the node-specificbeam sequence is defined by a sequence of power levels corresponding todistinct beam angles available to the access node. Note that when thearray comprises both horizontally and vertically spaced elements, eachavailable beam angle corresponds to a specific combination of azimuthand elevation; in embodiments where only horizontal elements are used,on the other hand, each available beam angle may be represented by onlya distinct azimuth. The method may be implemented in a distributedmanner, by the access nodes, in some embodiments, and in a centralizedmanner, e.g., by a control node or by some other node, in others.

As shown at block 710, the illustrated method comprises modeling a totalpower function corresponding to a total power transmitted in thesynchronization signals by the plurality of access nodes, for a givenperiod. This modeling of the total power function comprises modeling thetotal power function as a factor graph having a plurality of check nodesand variable nodes, each of the check nodes corresponding to one of aplurality of emulated virtual wireless devices in the region and each ofthe variable nodes corresponding to one of the access nodes and to oneof the beam angles available to the one of the access nodes.

As shown at block 720, the illustrated method further comprisesemulating the plurality of virtual wireless devices so as to implementquality-of-service constraints on synchronization signals received bythe virtual wireless devices. Further, as shown at block 730, the methodcomprises applying an iterative message-passing algorithm to the modeledtotal power function, to determine the sequence of power levels for eachof the plurality of access nodes, so as to minimize the total powerfunction, subject to one or more iteration-stopping criteria. In someembodiments of the illustrated method, synchronization signals aretransmitted from one or more of the access nodes, applying thedetermined sequences of non-zero power levels for the access node.

In some embodiments of the illustrated method, the iterativemessage-passing algorithm is a min-sum algorithm, as detailed above. Inother embodiments, the iterative message-passing algorithm may be amax-sum algorithm or a min-product algorithm, for example.

In some embodiments, the emulating of the plurality of virtual wirelessdevices is based on historical signal measurement data from wirelessdevices in the region, the historical signal measurement data comprisingreceived signal strength data for synchronization signals transmitted bythe plurality of access nodes. In some of these embodiments, at least aportion of the historical signal measurement data corresponds tomeasurements made from full power sweeps of the synchronization by oneor more of the access nodes.

In some embodiments, the method further comprises, subsequent to themodeling, emulating, and applying shown in blocks 710-730, updating thehistorical signal measurement data, using signal strength data fromwireless devices for synchronization signals transmitted by theplurality of access nodes using the determined sequences of non-zeropower levels. This is shown at block 740, which is shown with a dashedoutline to indicate that it need not appear in every embodiment orinstance of the illustrated method. After this updating, the modeling,emulating, and applying of blocks 710-730 are repeated to obtain updatedsequences of non-zero power levels for one or more of the access nodes.

In some embodiments, applying the iterative message-passing algorithm tothe modeled total power function includes generating, for each variablenode, an initial aggregate message targeted to each check node thatcorresponds to a virtual wireless device that, based on historicalsignal quality data, is deemed able to receive a synchronization signaltransmitted by the access node corresponding to the variable node at thecombination of azimuth and elevation beam angles corresponding to thevariable node, each of the initial aggregate messages consisting of avariable node power function that is a function of transmitted powerlevels for the variable node. This was shown at block 410 of FIG. 4. Theinitial aggregate messages are then passed to the respective checknodes, as shown at block 420 of FIG. 4.

As was illustrated at block 440, each check node then computes a summarymessage targeted to each of a set of variable nodes that, based onhistorical signal quality data, correspond to beams that can be receivedby the check node, each summary message being a function of transmitpower levels for the targeted variable node and having a value, for eachtransmit power of the targeted variable node, computed as the minimum,over all transmit power levels for all neighboring variable nodes exceptthe targeted variable node, of the sum of (i) an indicator function forthe check node and (ii) the aggregate messages most recently received bythe check node from all the variable nodes in the set other than thetargeted variable node, where the indicator function for the check nodeis a function of the transmit power level for the targeted variable nodeand provides a zero for a given transmit power level for the targetedvariable node if the signal quality for any of the beams that can bereceived by the check node, based on the transmit power level for thetargeted variable node and transmit power levels for the neighboringvariable nodes other than the target variable nodes, exceeds apredetermined threshold, and provides a predetermined penalty valueotherwise. The summary messages are then passed to the respectivevariable nodes. The signal qualities are determined/hypothesized fromthe historical statistics, combined with every possible power setting.The “feasible region/search space” is then determined by the indicatorfunction, which penalizes those power settings that do not lead tosignal qualities above the SINR threshold. To this end, at each checknode, every possible SINR value observed by that check node must behypothesized. “Beliefs” regarding the most recently determined powerlevels for the neighboring variable nodes are conveyed by aggregatemessages. The sum of incoming aggregate messages at a given check nodedefines a multi-variate belief. To compute summary messages, a givencheck node then evaluates the most recently determined multi-variatebelief at every “feasible” power setting, that is, the power settingsthat belong to the feasible region.

Next, a transmitted power level that minimizes the sum of the summarymessages most recently passed to the variable node is computed by eachvariable node—this was shown at block 460 of FIG. 4. Each variable nodethen computes an updated aggregate message targeted to each of the checknodes that corresponds to a virtual wireless device that, based onhistorical signal quality data, is deemed able to receive asynchronization signal transmitted by the access node corresponding tothe variable node at the combination of azimuth and elevation beamangles corresponding to the variable node, each of the updated aggregatemessages being computed as a sum of (iii) a power term that is afunction of transmitted power levels for the variable node and (iv) asum of the summary messages most recently passed to the variable nodefrom check nodes other than the targeted check node, less (v) theminimum of all the summary messages most recently passed to the variablenode from check nodes other than the targeted check node. The updatedaggregated messages are then passed to the respective check nodes. Thecomputing of the summary messages, the passing of the summary messages,the computing of the updated aggregated messages, and the passing of theupdated messages, as was illustrated in FIG. 4, until aniteration-stopping criterion is reached. The transmitted power levelsfor the variable nodes determined just prior to the iteration-stoppingcriterion represent the determined non-zero power levels for the beamsequences for the plurality of access nodes.

It will be appreciated that embodiments of the presently disclosedtechniques include a system for jointly determining beam-sweepingpatterns for synchronization signals transmitted in a region by each ofa plurality of access nodes in a wireless network, e.g., according tothe methods illustrated in FIGS. 4 and 7, as described above, and invariations thereof. Other embodiments include computer program productsthat include computer program instructions, for execution by one or morenodes in a wireless network, for carrying out one or more of suchmethods.

FIG. 8 illustrates an example access node 800 in which all or parts ofthe techniques described above may be implemented. The example accessnode 810, which is connected to an antenna array 840, comprises aprocessing circuit 810, a radio transceiver 820, and a network interfacecircuit 830. The radio transceiver 820, which is connected to theantenna array 840, is configured to receive signals from and transmitsignals to wireless devices, including synchronization signals asdiscussed herein. Processing circuit 810, which in the illustratedexample includes a processor 812 and a memory circuit 814, isconfigured, e.g., using program code stored in memory 814 for executionby processor 812, to control the radio transceiver 820 and the networkinterface circuit 830, as well as to carry out all or parts of thevarious methods described herein, including, for example, the methodsillustrated in FIGS. 4 and 7 and described above.

What is claimed is:
 1. A method, in one or more nodes of a wirelesscommunications system, of jointly determining beam-sweeping patterns forsynchronization signals transmitted in a region by each of a pluralityof access nodes in a wireless network, wherein each access node isconnected to a corresponding array of horizontally and/or verticallyspaced antenna elements and is configured to sweep a synchronizationsignal in a node-specific beam sequence, using the corresponding array,where, for each access node, the node-specific beam sequence is definedby a sequence of power levels corresponding to distinct beam anglesavailable to the access node, the method comprising: modeling a totalpower function corresponding to a total power transmitted in thesynchronization signals by the plurality of access nodes, for a givenperiod, wherein said modeling of the total power function comprisesmodeling the total power function as a factor graph having a pluralityof check nodes and variable nodes, each of the check nodes correspondingto one of a plurality of emulated virtual wireless devices in the regionand each of the variable nodes corresponding to one of the access nodesand to one of the beam angles available to the one of the access nodes;emulating the plurality of virtual wireless devices so as to implementquality-of-service constraints on synchronization signals received bythe virtual wireless devices; applying an iterative message-passingalgorithm to the modeled total power function, to determine the sequenceof power levels for each of the plurality of access nodes, so as tominimize the total power function, subject to one or moreiteration-stopping criteria.
 2. The method of claim 1, wherein theiterative message-passing algorithm is one of a min-sum algorithm, amax-sum algorithm, and a min-product algorithm.
 3. The method of claim1, further comprising transmitting the synchronization signal from oneof the access nodes, applying the determined sequences of non-zero powerlevels for the access node.
 4. The method of claim 1, wherein theemulating of the plurality of virtual wireless devices is based onhistorical signal measurement data from wireless devices in the region,the historical signal measurement data comprising received signalstrength data for synchronization signals transmitted by the pluralityof access nodes.
 5. The method of claim 4, wherein at least a portion ofthe historical signal measurement data corresponds to measurements madefrom full power sweeps of the synchronization by one or more of theaccess nodes.
 6. The method of claim 4, wherein the method furthercomprises, subsequent to said modeling, emulating, and applying:updating the historical signal measurement data, using signal strengthdata from wireless devices for synchronization signals transmitted bythe plurality of access nodes using the determined sequences of non-zeropower levels; and repeating said modeling, emulating, and applying toobtain updated sequences of non-zero power levels for one or more of theaccess nodes.
 7. The method of claim 1, wherein said applying theiterative message-passing algorithm to the modeled total power functioncomprises: generating, for each variable node, an initial aggregatemessage targeted to each check node that corresponds to a virtualwireless device that, based on historical signal quality data, is deemedable to receive a synchronization signal transmitted by the access nodecorresponding to the variable node at the combination of azimuth andelevation beam angles corresponding to the variable node, each of theinitial aggregate messages consisting of a variable node power functionthat is a function of transmit power levels for the variable node;passing the initial aggregate messages to the respective check nodes;computing, for each check node, a summary message targeted to each of aset of variable nodes that, based on historical signal quality data,correspond to beams that can be received by the check node, each summarymessage being a function of transmit power levels for the targetedvariable node and having a value, for each transmit power of thetargeted variable node, computed as the minimum, over all transmit powerlevels for all neighboring variable nodes except the targeted variablenode, of the sum of (i) an indicator function for the check node and(ii) the aggregate messages most recently received by the check nodefrom all the variable nodes in the set other than the targeted variablenode, wherein the indicator function for the check node is a function ofthe transmit power level for the targeted variable node and provides azero for a given transmit power level for the targeted variable node ifthe signal quality for any of the beams that can be received by thecheck node, as hypothesized based on the transmit power level for thetargeted variable node and transmit power levels for the neighboringvariable nodes other than the target variable nodes, exceeds apredetermined threshold, and provides a predetermined penalty valueotherwise; passing the summary messages to the respective variablenodes; determining, for each variable node, a transmit power level thatminimizes the sum of the summary messages most recently passed to thevariable node; computing, for each variable node, an updated aggregatemessage targeted to each of the check nodes that corresponds to avirtual wireless device that, based on historical signal quality data,is deemed able to receive a synchronization signal transmitted by theaccess node corresponding to the variable node at the combination ofazimuth and elevation beam angles corresponding to the variable node,each of the updated aggregate messages being computed as a sum of (iii)a power term that is a function of transmit power levels for thevariable node and (iv) a sum of the summary messages most recentlypassed to the variable node from check nodes other than the targetedcheck node, less (v) the minimum of all the summary messages mostrecently passed to the variable node from check nodes other than thetargeted check node; passing the updated aggregated messages to therespective check nodes; repeating the computing of the summary messages,the passing of the summary messages, the computing of the updatedaggregated messages, and the passing of the updated messages, until aniteration-stopping criterion is reached; and wherein the transmit powerlevels for the variable nodes determined just prior to theiteration-stopping criterion represent the determined non-zero powerlevels for the beam sequences for the plurality of access nodes.
 8. Themethod of claim 1, wherein emulating the plurality of virtual wirelessdevices so as to implement quality-of-service constraints onsynchronization signals received by the virtual wireless devices iscarried out so as to ignore interference.
 9. The method of claim 8,wherein said applying the iterative message-passing algorithm to themodeled total power function comprises: generating, for each variablenode, an initial aggregate message targeted to each check node thatcorresponds to a virtual wireless device that, based on historicalsignal quality data, is deemed able to receive a synchronization signaltransmitted by the access node corresponding to the variable node at thecombination of azimuth and elevation beam angles corresponding to thevariable node, each of the initial aggregate messages consisting of avariable node power function that is a function of transmit power levelsfor the variable node; passing the initial aggregate messages to therespective check nodes; computing, for each check node, a summarymessage targeted to each of a set of variable nodes that, based onhistorical signal quality data, correspond to beams that can be receivedby the check node, each summary message being computed as (i) a zero ifthe beam power for the variable node exceeds a minimum power levelneeded for the corresponding beam to serve the virtual nodecorresponding to the check node and, otherwise, (b) as the minimum, overall the aggregate messages most recently received by the check node fromall the variable nodes in the set other than the targeted variable node,of the values of the aggregate messages in ranges of the aggregatemessages in which the transmit power of the corresponding variable nodeexceeds a minimum power level needed for the corresponding variable nodeto serve the virtual node corresponding to the check node; passing thesummary messages to the respective variable nodes; determining, for eachvariable node, a transmit power level that minimizes the sum of thesummary messages most recently passed to the variable node; computing,for each variable node, an updated aggregate message targeted to each ofthe check nodes that corresponds to a virtual wireless device that,based on historical signal quality data, is deemed able to receive asynchronization signal transmitted by the access node corresponding tothe variable node at the combination of azimuth and elevation beamangles corresponding to the variable node, each of the updated aggregatemessages being computed as a sum of (iii) a power term that is afunction of transmit power levels for the variable node and (iv) a sumof the summary messages most recently passed to the variable node fromcheck nodes other than the targeted check node, less (v) the minimum ofall the summary messages most recently passed to the variable node fromcheck nodes other than the targeted check node; passing the updatedaggregated messages to the respective check nodes; repeating thecomputing of the summary messages, the passing of the summary messages,the computing of the updated aggregated messages, and the passing of theupdated messages, until an iteration-stopping criterion is reached; andwherein the transmitted power levels for the variable nodes determinedjust prior to the iteration-stopping criterion represent the determinednon-zero power levels for the beam sequences for the plurality of accessnodes.
 10. The method of claim 1, wherein the method is implemented in adistributed manner, in the plurality of access nodes, each access nodecomputing messages for the variable nodes corresponding to therespective node.
 11. The method of claim 10, wherein each of one or moreof the access nodes performs the emulation of a corresponding subset ofthe plurality of wireless devices.
 12. A system for jointly determiningbeam-sweeping patterns for synchronization signals transmitted in aregion by each of a plurality of access nodes in a wireless network,wherein each access node is connected to a corresponding array ofhorizontally and/or vertically spaced antenna elements and is configuredto sweep a synchronization signal in a node-specific beam sequence,using the corresponding array, where, for each access node, thenode-specific beam sequence is defined by a sequence of power levelscorresponding to distinct beam angles available to the access node, thesystem comprising one or more nodes adapted to: model a total powerfunction corresponding to a total power transmitted in thesynchronization signals by the plurality of access nodes, for a givenperiod, wherein said modeling of the total power function comprisesmodeling the total power function as a factor graph having a pluralityof check nodes and variable nodes, each of the check nodes correspondingto one of a plurality of emulated virtual wireless devices in the regionand each of the variable nodes corresponding to one of the access nodesand to one of the beam angles available to the one of the access nodes;emulate the plurality of virtual wireless devices so as to implementquality-of-service constraints on synchronization signals received bythe virtual wireless devices; apply an iterative message-passingalgorithm to the modeled total power function, to determine the sequenceof power levels for each of the plurality of access nodes, so as tominimize the total power function, subject to one or moreiteration-stopping criteria.
 13. The system of claim 12, wherein theiterative message-passing algorithm is one of a min-sum algorithm, amax-sum algorithm, and a min-product algorithm.
 14. The system of claim12, wherein at least a first one of the nodes is an access node and isfurther adapted to transmit the synchronization signal, applying thedetermined sequences of non-zero power levels for that access node. 15.The system of claim 12, wherein the nodes are adapted to emulate theplurality of virtual wireless devices based on historical signalmeasurement data from wireless devices in the region, the historicalsignal measurement data comprising received signal strength data forsynchronization signals transmitted by the plurality of access nodes.16. The system of claim 15, wherein at least a portion of the historicalsignal measurement data corresponds to measurements made from full powersweeps of the synchronization by one or more of the access nodes. 17.The system of claim 15, wherein the nodes are further adapted to,subsequent to said modeling, emulating, and applying: update thehistorical signal measurement data, using signal strength data fromwireless devices for synchronization signals transmitted by theplurality of access nodes using the determined sequences of non-zeropower levels; and repeat said modeling, emulating, and applying toobtain updated sequences of non-zero power levels for one or more of theaccess nodes.
 18. The system of claim 12, wherein the nodes are adaptedto apply the iterative message-passing algorithm to the modeled totalpower function by: generating, for each variable node, an initialaggregate message targeted to each check node that corresponds to avirtual wireless device that, based on historical signal quality data,is deemed able to receive a synchronization signal transmitted by theaccess node corresponding to the variable node at the combination ofazimuth and elevation beam angles corresponding to the variable node,each of the initial aggregate messages consisting of a variable nodepower function that is a function of transmit power levels for thevariable node; passing the initial aggregate messages to the respectivecheck nodes; computing, for each check node, a summary message targetedto each of a set of variable nodes that, based on historical signalquality data, correspond to beams that can be received by the checknode, each summary message being a function of transmit power levels forthe targeted variable node and having a value, for each transmit powerof the targeted variable node, computed as the minimum, over alltransmit power levels for all neighboring variable nodes except thetargeted variable node, of the sum of (i) an indicator function for thecheck node and (ii) the aggregate messages most recently received by thecheck node from all the variable nodes in the set other than thetargeted variable node, wherein the indicator function for the checknode is a function of the transmit power level for the targeted variablenode and provides a zero for a given transmit power level for thetargeted variable node if the signal quality for any of the beams thatcan be received by the check node, as hypothesized based on the transmitpower level for the targeted variable node and transmit power levels forthe neighboring variable nodes other than the target variable nodes,exceeds a predetermined threshold, and provides a predetermined penaltyvalue otherwise; passing the summary messages to the respective variablenodes; determining, for each variable node, a transmit power level thatminimizes the sum of the summary messages most recently passed to thevariable node; computing, for each variable node, an updated aggregatemessage targeted to each of the check nodes that corresponds to avirtual wireless device that, based on historical signal quality data,is deemed able to receive a synchronization signal transmitted by theaccess node corresponding to the variable node at the combination ofazimuth and elevation beam angles corresponding to the variable node,each of the updated aggregate messages being computed as a sum of (iii)a power term that is a function of transmit power levels for thevariable node and (iv) a sum of the summary messages most recentlypassed to the variable node from check nodes other than the targetedcheck node, less (v) the minimum of all the summary messages mostrecently passed to the variable node from check nodes other than thetargeted check node; passing the updated aggregated messages to therespective check nodes; repeating the computing of the summary messages,the passing of the summary messages, the computing of the updatedaggregated messages, and the passing of the updated messages, until aniteration-stopping criterion is reached; and wherein the transmit powerlevels for the variable nodes determined just prior to theiteration-stopping criterion represent the determined non-zero powerlevels for the beam sequences for the plurality of access nodes.
 19. Thesystem of claim 12, wherein the nodes are adapted to emulate theplurality of virtual wireless devices such that the emulating ignoresinterference.
 20. The system of claim 19, wherein the nodes are adaptedto apply the iterative message-passing algorithm to the modeled totalpower function by: generating, for each variable node, an initialaggregate message targeted to each check node that corresponds to avirtual wireless device that, based on historical signal quality data,is deemed able to receive a synchronization signal transmitted by theaccess node corresponding to the variable node at the combination ofazimuth and elevation beam angles corresponding to the variable node,each of the initial aggregate messages consisting of a variable nodepower function that is a function of transmitted power levels for thevariable node; passing the initial aggregate messages to the respectivecheck nodes; computing, for each check node, a summary message targetedto each of a set of variable nodes that, based on historical signalquality data, correspond to beams that can be received by the checknode, each summary message being computed as (i) a zero if the beampower for the variable node exceeds a minimum power level needed for thecorresponding beam to serve the virtual node corresponding to the checknode and, otherwise, (b) as the minimum, over all the aggregate messagesmost recently received by the check node from all the variable nodes inthe set other than the targeted variable node, of the values of theaggregate messages in ranges of the aggregate messages in which thetransmit power of the corresponding variable node exceeds a minimumpower level needed for the corresponding variable node to serve thevirtual node corresponding to the check node; passing the summarymessages to the respective variable nodes; determining, for eachvariable node, a transmitted power level that minimizes the sum of thesummary messages most recently passed to the variable node; computing,for each variable node, an updated aggregate message targeted to each ofthe check nodes that corresponds to a virtual wireless device that,based on historical signal quality data, is deemed able to receive asynchronization signal transmitted by the access node corresponding tothe variable node at the combination of azimuth and elevation beamangles corresponding to the variable node, each of the updated aggregatemessages being computed as a sum of (iii) a power term that is afunction of transmitted power levels for the variable node and (iv) asum of the summary messages most recently passed to the variable nodefrom check nodes other than the targeted check node, less (v) theminimum of all the summary messages most recently passed to the variablenode from check nodes other than the targeted check node; passing theupdated aggregated messages to the respective check nodes; repeating thecomputing of the summary messages, the passing of the summary messages,the computing of the updated aggregated messages, and the passing of theupdated messages, until an iteration-stopping criterion is reached; andwherein the transmitted power levels for the variable nodes determinedjust prior to the iteration-stopping criterion represent the determinednon-zero power levels for the beam sequences for the plurality of accessnodes.
 21. The system of claim 12, wherein each of one or more of thenodes corresponds to a respective one of the access nodes, and whereineach of the one or more of the nodes performs the emulation of acorresponding subset of the plurality of wireless devices.
 22. Acomputer program product, stored on a non-transitory computer readablestorage medium, for jointly determining beam-sweeping patterns forsynchronization signals transmitted in a region by each of a pluralityof access nodes in a wireless network, wherein each access node isconnected to a corresponding array of horizontally and/or verticallyspaced antenna elements and is configured to sweep a synchronizationsignal in a node-specific beam sequence, using the corresponding array,where, for each access node, the node-specific beam sequence is definedby a sequence of power levels corresponding to distinct beam anglesavailable to the access node, the computer program product comprisinginstructions for execution by one or more nodes, the instructionscomprising instructions for: modeling a total power functioncorresponding to a total power transmitted in the synchronizationsignals by the plurality of access nodes, for a given period, whereinsaid modeling of the total power function comprises modeling the totalpower function as a factor graph having a plurality of check nodes andvariable nodes, each of the check nodes corresponding to one of aplurality of emulated virtual wireless devices in the region and each ofthe variable nodes corresponding to one of the access nodes and to oneof the beam angles available to the one of the access nodes; emulatingthe plurality of virtual wireless devices so as to implementquality-of-service constraints on synchronization signals received bythe virtual wireless devices; applying an iterative message-passingalgorithm to the modeled total power function, to determine the sequenceof power levels for each of the plurality of access nodes, so as tominimize the total power function, subject to one or moreiteration-stopping criteria.